Belinfante–Rosenfeld stress–energy tensor
In mathematical physics, the Belinfante–Rosenfeld tensor is a modification of the energy–momentum tensor that is constructed from the canonical energy–momentum tensor and the spin current so as to be symmetric yet still conserved.
of a local current
Thus a source of spin-current implies a non-symmetric canonical energy–momentum tensor.
that is constructed from the canonical energy momentum tensor and the spin current so as to be symmetric yet still conserved.
An integration by parts shows that
and so a physical interpretation of Belinfante tensor is that it includes the "bound momentum" associated with gradients of the intrinsic angular momentum. In other words, the added term is an analogue of the "bound current" associated with a magnetization density .
The curious combination of spin-current components required to make symmetric and yet still conserved seems totally ad hoc, but it was shown by both Rosenfeld and Belinfante that the modified tensor is precisely the symmetric Hilbert energy–momentum tensor that acts as the source of gravity in general relativity. Just as it is the sum of the bound and free currents that acts as a source of the magnetic field, it is the sum of the bound and free energy–momentum that acts as a source of gravity.
- F. J. Belinfante (1940). "On the current and the density of the electric charge, the energy, the linear momentum and the angular momentum of arbitrary fields". Physica 7: 449. doi:10.1016/S0031-8914(40)90091-X.
- L. Rosenfeld (1940). "Sur le tenseur D’Impulsion- Energie". Acad. Roy. Belg. Memoirs de classes de Science 18.